His initial explanation of uncountable infinity isn't exactly correct, and I'm afraid it will give people the wrong idea. He says that the real numbers are uncountable, because even between 0 and 1 on the number line there are an infinite number of real numbers. But that is also true of the rational numbers, which are countable! After all, what is the smallest rational number larger than 0?
at each step you list the numbers where the numerator and denominator are <= x. For example, if x = 2, we can count 1, -1, 2, -2, 1/2, and -1/2. Obviously it is an infinite list, but you can list them.
